We address the problem of detection and estimation of one or two change-points in the mean of a series of random variables. We use the formalism of set estimation in regression: to each point of a design is attached a binary label that indicates whether that point belongs to an unknown segment and this label is contaminated with noise. The endpoints of the unknown segment are the change-points. We study the minimal size of the segment which allows statistical detection in different scenarios, including when the endpoints are separated from the boundary of the domain of the design, or when they are separated from one another. We compare this minimal size with the minimax rates of convergence for estimation of the segment under the same scenarios. The aim of this extensive study of a simple yet fundamental version of the change-point problem is two-fold: understanding the impact of the location and the separation of the change points on detection and estimation and bringing insights about the estimation and detection of convex bodies in higher dimensions.
DOI : 10.1051/ps/2018014
Mots-clés : Change-point, detection, hypothesis testing, minimax, separation rate, set estimation
@article{PS_2018__22__210_0,
author = {Brunel, Victor-Emmanuel},
title = {A change-point problem and inference for segment signals},
journal = {ESAIM: Probability and Statistics},
pages = {210--235},
publisher = {EDP-Sciences},
volume = {22},
year = {2018},
doi = {10.1051/ps/2018014},
mrnumber = {3891756},
zbl = {1409.62165},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2018014/}
}
TY - JOUR AU - Brunel, Victor-Emmanuel TI - A change-point problem and inference for segment signals JO - ESAIM: Probability and Statistics PY - 2018 SP - 210 EP - 235 VL - 22 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2018014/ DO - 10.1051/ps/2018014 LA - en ID - PS_2018__22__210_0 ER -
Brunel, Victor-Emmanuel. A change-point problem and inference for segment signals. ESAIM: Probability and Statistics, Tome 22 (2018), pp. 210-235. doi : 10.1051/ps/2018014. http://www.numdam.org/articles/10.1051/ps/2018014/
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